From 332976573fe1cd261346830f81ccc86a49f8ba9b Mon Sep 17 00:00:00 2001 From: Per Alexandersson Date: Thu, 22 Oct 2026 09:36:06 +0000 Subject: [PATCH] =?UTF-8?q?Ferrers=20tilings=20(Ahlberg=E2=80=93Alexanders?= =?UTF-8?q?son),=20layer=201:=20real-rootedness=20results?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Worker-drafted (Codex), integrated unverified: polynomial families from arXiv:2605.03473 defined by recurrence / closed form, with the paper's real-rootedness and interlacing results (two-column and fault-line families complete; L-tile, B_d, rigid and row-refined partial). The combinatorial model (layer 2) belongs in AlexanderssonFormalizations. Co-Authored-By: Claude Opus 5.5 --- RealRooted.lean | 8 + RealRooted/Challenges/FerrersTilings.lean | 174 +++++++++++++ RealRooted/FerrersTiling/AnchorRefined.lean | 242 ++++++++++++++++++ .../FerrersTiling/FaultFreeRectangle.lean | 96 +++++++ RealRooted/FerrersTiling/FaultLines.lean | 89 +++++++ RealRooted/FerrersTiling/LTiling.lean | 74 ++++++ RealRooted/FerrersTiling/Rigid.lean | 69 +++++ RealRooted/FerrersTiling/RowRefined.lean | 45 ++++ RealRooted/FerrersTiling/TwoColumn.lean | 98 +++++++ RealRooted/Production.lean | 8 + scripts/import_architecture.json | 4 +- website/symcat/bibliography.json | 26 ++ website/symcat/labels.json | 2 +- 13 files changed, 932 insertions(+), 3 deletions(-) create mode 100644 RealRooted/Challenges/FerrersTilings.lean create mode 100644 RealRooted/FerrersTiling/AnchorRefined.lean create mode 100644 RealRooted/FerrersTiling/FaultFreeRectangle.lean create mode 100644 RealRooted/FerrersTiling/FaultLines.lean create mode 100644 RealRooted/FerrersTiling/LTiling.lean create mode 100644 RealRooted/FerrersTiling/Rigid.lean create mode 100644 RealRooted/FerrersTiling/RowRefined.lean create mode 100644 RealRooted/FerrersTiling/TwoColumn.lean diff --git a/RealRooted.lean b/RealRooted.lean index f106efa4b..ca97c46a3 100644 --- a/RealRooted.lean +++ b/RealRooted.lean @@ -2062,4 +2062,12 @@ import RealRooted.Mathlib.Algebra.LinearRecurrence.Operator import RealRooted.RecurrenceSpace.Defect import RealRooted.RecurrenceSpace.Filtered import RealRooted.RecurrenceSpace.Operators +import RealRooted.FerrersTiling.AnchorRefined +import RealRooted.FerrersTiling.FaultFreeRectangle +import RealRooted.FerrersTiling.FaultLines +import RealRooted.FerrersTiling.LTiling +import RealRooted.FerrersTiling.Rigid +import RealRooted.FerrersTiling.RowRefined +import RealRooted.FerrersTiling.TwoColumn +import RealRooted.Challenges.FerrersTilings import RealRooted.Tactic.Examples.A049325RecurrenceSpace diff --git a/RealRooted/Challenges/FerrersTilings.lean b/RealRooted/Challenges/FerrersTilings.lean new file mode 100644 index 000000000..58907db7f --- /dev/null +++ b/RealRooted/Challenges/FerrersTilings.lean @@ -0,0 +1,174 @@ +import RealRooted.FerrersTiling.AnchorRefined +import RealRooted.FerrersTiling.FaultFreeRectangle +import RealRooted.FerrersTiling.FaultLines +import RealRooted.FerrersTiling.LTiling +import RealRooted.FerrersTiling.RowRefined + +/-! +# Algebraic Ferrers-tiling polynomial families + + + + +# Polynomials from tilings of rectangles + +Ahlberg and Alexandersson associate several polynomial families with tilings +of rectangles by Ferrers shapes. At this algebraic layer we define the +families by the recurrences and coefficient formulas from the paper; the +tiling bijections are separate combinatorial statements. + +The terminal-state-refined anchor polynomials obey the paper's transfer +matrix: the row for $\infty$ consists of ones, while row $j$ consists of $t$ +in the columns before $j$ and zeroes thereafter. The library's zero-aware +matrix-preserver theorem proves that every resulting vector is an interlacing +sequence [@AhlbergAlexandersson2026, Thm. 3.5]. + +For a two-column tile and parameter $d$, the polynomial of width $n$ is + +$$ +P_n(t)=\sum_j \binom{n-j}{j}d^j t^j. +$$ + +It satisfies $P_{n+2}=P_{n+1}+dtP_n$, is real-rooted, and satisfies +$P_n\preceq P_{n+1}$ [@AhlbergAlexandersson2026, Thm. 4.1]. Reversing its +coefficients gives the fault-line family +$R_{n+2}=tR_{n+1}+dtR_n$, which is also real-rooted. + +For a rigid tile of width $\ell+1$, the polynomial is + +$$ +P_{d,\ell,n}(t)=\sum_j \binom{d(n-j\ell)}{j}t^j. +$$ + +This is the path-power polynomial +`pathPowerCliquePolynomial (d * ell + 1) (d * n)`. Powers of paths are +claw-free, so the Chudnovsky--Seymour theorem makes the polynomial +Pólya-frequency; in particular all its zeros are real and nonpositive +[@AhlbergAlexandersson2026, Thm. 5.3]. + +The fault-free rectangle polynomial is another path-power polynomial. Its +closed coefficients give the paper's bars-and-stars formula, its path +deletion recurrence is $B_d=B_{d-1}+tB_{d-k}$, and consecutive members +interlace [@AhlbergAlexandersson2026, Lem. 6.3]. Finally, the univariate +row-refined specialization + +$$ +\sum_j \binom{n-j}{j}(s+d-1)^j +$$ + +is a translate of a path-power polynomial and is therefore real-rooted +[@AhlbergAlexandersson2026, Thm. 7.1]. + +## Proof idea + +All four families are identified with independence polynomials of powers of +paths, possibly after a positive scaling, translation, or coefficient +reversal. The library's path-power theory supplies both real-rootedness and +interlacing; reflection preserves splitting. + +## References + +The families and formulas are from [@AhlbergAlexandersson2026]. The +real-rootedness engine for path-power independence polynomials is the +claw-free independence-polynomial theorem of [@ChudnovskySeymour2007]. + +-/ + +namespace RealRooted.Challenges.FerrersTilings + +end RealRooted.Challenges.FerrersTilings diff --git a/RealRooted/FerrersTiling/AnchorRefined.lean b/RealRooted/FerrersTiling/AnchorRefined.lean new file mode 100644 index 000000000..28a803109 --- /dev/null +++ b/RealRooted/FerrersTiling/AnchorRefined.lean @@ -0,0 +1,242 @@ +import RealRooted.MatrixInterlacing + +/-! +# Refined anchor-word recursion + +This file formalizes the polynomial transfer matrix from +Ahlberg--Alexandersson, *Polynomials from tilings of rectangles*, Section 3. +Index `0` represents the terminal symbol `∞`; index `j + 1` represents row +`j + 1`. The construction is algebraic: the layer-2 anchor-word model is not +used here. +-/ + +open Polynomial + +namespace RealRooted +namespace FerrersTiling + +noncomputable section + +/-- Entry `(i,j)` of the refined anchor transfer matrix. Its first row is +all `1`; row `i > 0` contains `X` before column `i` and `0` thereafter. -/ +def anchorTransferEntry (i j : ℕ) : ℝ[X] := + if i = 0 then 1 else if j < i then X else 0 + +/-- The `(d+1) × (d+1)` transfer matrix for the refined polynomials +`(Q_(d,n,∞), Q_(d,n,1), ..., Q_(d,n,d))`. -/ +def anchorTransferMatrix (d : ℕ) : List (List ℝ[X]) := + (List.range (d + 1)).map fun i => + (List.range (d + 1)).map fun j => anchorTransferEntry i j + +@[simp] theorem length_anchorTransferMatrix (d : ℕ) : + (anchorTransferMatrix d).length = d + 1 := by + simp [anchorTransferMatrix] + +theorem anchorTransferMatrix_rect (d : ℕ) : + ∀ row ∈ anchorTransferMatrix d, row.length = d + 1 := by + intro row hrow + simp only [anchorTransferMatrix, List.mem_map, List.mem_range] at hrow + obtain ⟨i, hi, rfl⟩ := hrow + simp + +@[simp] theorem getElem_anchorTransferMatrix (d i j : ℕ) + (hi : i < (anchorTransferMatrix d).length) + (hj : j < (anchorTransferMatrix d)[i].length) : + (anchorTransferMatrix d)[i][j] = anchorTransferEntry i j := by + simp [anchorTransferMatrix] + +theorem anchorTransferMatrix_nonneg (d : ℕ) : + ∀ row ∈ anchorTransferMatrix d, ∀ p ∈ row, HasNonnegCoeffs p := by + intro row hrow p hp + simp only [anchorTransferMatrix, List.mem_map, List.mem_range] at hrow + obtain ⟨i, hi, rfl⟩ := hrow + simp only [List.mem_map, List.mem_range] at hp + obtain ⟨j, hj, rfl⟩ := hp + unfold anchorTransferEntry + split_ifs + · exact hasNonnegCoeffs_one + · exact hasNonnegCoeffs_X + · exact hasNonnegCoeffs_zero + +private theorem atomic_has2x2_1111 : + Has2x2InterlacingProperty0 1 1 1 1 := by + exact Atomic2x2Matrix.hasAffineProperty_of_isAffineAdmissible + (M := ⟨.one, .one, .one, .one⟩) (by decide) + +private theorem atomic_has2x2_11XX : + Has2x2InterlacingProperty0 1 1 X X := by + exact Atomic2x2Matrix.hasAffineProperty_of_isAffineAdmissible + (M := ⟨.one, .one, .x, .x⟩) (by decide) + +private theorem atomic_has2x2_11X0 : + Has2x2InterlacingProperty0 1 1 X 0 := by + exact Atomic2x2Matrix.hasAffineProperty_of_isAffineAdmissible + (M := ⟨.one, .one, .x, .zero⟩) (by decide) + +private theorem atomic_has2x2_1100 : + Has2x2InterlacingProperty0 1 1 0 0 := by + exact Atomic2x2Matrix.hasAffineProperty_of_isAffineAdmissible + (M := ⟨.one, .one, .zero, .zero⟩) (by decide) + +private theorem atomic_has2x2_XXXX : + Has2x2InterlacingProperty0 X X X X := by + exact Atomic2x2Matrix.hasAffineProperty_of_isAffineAdmissible + (M := ⟨.x, .x, .x, .x⟩) (by decide) + +private theorem atomic_has2x2_X0XX : + Has2x2InterlacingProperty0 X 0 X X := by + exact Atomic2x2Matrix.hasAffineProperty_of_isAffineAdmissible + (M := ⟨.x, .zero, .x, .x⟩) (by decide) + +private theorem atomic_has2x2_X0X0 : + Has2x2InterlacingProperty0 X 0 X 0 := by + exact Atomic2x2Matrix.hasAffineProperty_of_isAffineAdmissible + (M := ⟨.x, .zero, .x, .zero⟩) (by decide) + +private theorem atomic_has2x2_00XX : + Has2x2InterlacingProperty0 0 0 X X := by + exact Atomic2x2Matrix.hasAffineProperty_of_isAffineAdmissible + (M := ⟨.zero, .zero, .x, .x⟩) (by decide) + +private theorem atomic_has2x2_00X0 : + Has2x2InterlacingProperty0 0 0 X 0 := by + exact Atomic2x2Matrix.hasAffineProperty_of_isAffineAdmissible + (M := ⟨.zero, .zero, .x, .zero⟩) (by decide) + +private theorem atomic_has2x2_0000 : + Has2x2InterlacingProperty0 0 0 0 0 := by + exact Atomic2x2Matrix.hasAffineProperty_of_isAffineAdmissible + (M := ⟨.zero, .zero, .zero, .zero⟩) (by decide) + +private theorem anchorTransferEntry_has2x2 {i₁ i₂ j₁ j₂ : ℕ} + (hi : i₁ ≤ i₂) (hj : j₁ ≤ j₂) : + Has2x2InterlacingProperty0 + (anchorTransferEntry i₁ j₁) (anchorTransferEntry i₁ j₂) + (anchorTransferEntry i₂ j₁) (anchorTransferEntry i₂ j₂) := by + by_cases hi₁ : i₁ = 0 + · subst i₁ + by_cases hi₂ : i₂ = 0 + · subst i₂ + simpa [anchorTransferEntry] using atomic_has2x2_1111 + · have hi₂pos : 0 < i₂ := Nat.pos_of_ne_zero hi₂ + by_cases hj₂ : j₂ < i₂ + · have hj₁ : j₁ < i₂ := lt_of_le_of_lt hj hj₂ + simpa [anchorTransferEntry, hi₂, hj₁, hj₂] using atomic_has2x2_11XX + · by_cases hj₁ : j₁ < i₂ + · simpa [anchorTransferEntry, hi₂, hj₁, hj₂] using atomic_has2x2_11X0 + · simpa [anchorTransferEntry, hi₂, hj₁, hj₂] using atomic_has2x2_1100 + · have hi₁pos : 0 < i₁ := Nat.pos_of_ne_zero hi₁ + have hi₂pos : 0 < i₂ := hi₁pos.trans_le hi + have hi₂ : i₂ ≠ 0 := Nat.ne_of_gt hi₂pos + by_cases hj₂i₁ : j₂ < i₁ + · have hj₁i₁ : j₁ < i₁ := lt_of_le_of_lt hj hj₂i₁ + have hj₁i₂ : j₁ < i₂ := hj₁i₁.trans_le hi + have hj₂i₂ : j₂ < i₂ := hj₂i₁.trans_le hi + simpa [anchorTransferEntry, hi₁, hi₂, hj₁i₁, hj₂i₁, hj₁i₂, hj₂i₂] using + atomic_has2x2_XXXX + · by_cases hj₁i₁ : j₁ < i₁ + · have hj₁i₂ : j₁ < i₂ := hj₁i₁.trans_le hi + by_cases hj₂i₂ : j₂ < i₂ + · simpa [anchorTransferEntry, hi₁, hi₂, hj₁i₁, hj₂i₁, hj₁i₂, hj₂i₂] using + atomic_has2x2_X0XX + · simpa [anchorTransferEntry, hi₁, hi₂, hj₁i₁, hj₂i₁, hj₁i₂, hj₂i₂] using + atomic_has2x2_X0X0 + · by_cases hj₂i₂ : j₂ < i₂ + · have hj₁i₂ : j₁ < i₂ := lt_of_le_of_lt hj hj₂i₂ + simpa [anchorTransferEntry, hi₁, hi₂, hj₁i₁, hj₂i₁, hj₁i₂, hj₂i₂] using + atomic_has2x2_00XX + · by_cases hj₁i₂ : j₁ < i₂ + · simpa [anchorTransferEntry, hi₁, hi₂, hj₁i₁, hj₂i₁, hj₁i₂, hj₂i₂] using + atomic_has2x2_00X0 + · simpa [anchorTransferEntry, hi₁, hi₂, hj₁i₁, hj₂i₁, hj₁i₂, hj₂i₂] using + atomic_has2x2_0000 + +theorem anchorTransferMatrix_has2x2 (d : ℕ) : + ∀ (i₁ i₂ : Fin (anchorTransferMatrix d).length) (j₁ j₂ : Fin (d + 1)), + i₁ ≤ i₂ → j₁ ≤ j₂ → + Has2x2InterlacingProperty0 + (((anchorTransferMatrix d).get i₁).get + ⟨j₁, by rw [anchorTransferMatrix_rect d _ (List.get_mem _ i₁)]; exact j₁.2⟩) + (((anchorTransferMatrix d).get i₁).get + ⟨j₂, by rw [anchorTransferMatrix_rect d _ (List.get_mem _ i₁)]; exact j₂.2⟩) + (((anchorTransferMatrix d).get i₂).get + ⟨j₁, by rw [anchorTransferMatrix_rect d _ (List.get_mem _ i₂)]; exact j₁.2⟩) + (((anchorTransferMatrix d).get i₂).get + ⟨j₂, by rw [anchorTransferMatrix_rect d _ (List.get_mem _ i₂)]; exact j₂.2⟩) := by + intro i₁ i₂ j₁ j₂ hi hj + simp only [List.get_eq_getElem, getElem_anchorTransferMatrix] + exact anchorTransferEntry_has2x2 hi hj + +/-- The refined vector, initialized by `(1,0,...,0)` and advanced by the +paper's transfer matrix. -/ +def anchorRefinedList (d : ℕ) : ℕ → List ℝ[X] + | 0 => oneSupportSeq (d + 1) ⟨0, by lia⟩ + | n + 1 => matPolyAction (anchorTransferMatrix d) (anchorRefinedList d n) + +@[simp] theorem length_anchorRefinedList (d n : ℕ) : + (anchorRefinedList d n).length = d + 1 := by + cases n <;> simp [anchorRefinedList] + +/-- The exact vector recursion of the paper. -/ +theorem anchorRefinedList_succ (d n : ℕ) : + anchorRefinedList d (n + 1) = + matPolyAction (anchorTransferMatrix d) (anchorRefinedList d n) := by + rfl + +/-- The refined polynomial with terminal state `i`; state `0` is `∞`, and +state `j+1` is row `j+1`. -/ +def anchorRefinedPolynomial (d n : ℕ) (i : Fin (d + 1)) : ℝ[X] := + (anchorRefinedList d n).get ⟨i, by rw [length_anchorRefinedList]; exact i.2⟩ + +/-- Every refined vector is a zero-aware nonnegative interlacing sequence, +and every nonzero member is real-rooted. -/ +theorem anchorRefined_interlacing (d n : ℕ) : + IsInterlacingSeq0NonnegRealRooted (anchorRefinedList d n) := by + induction n with + | zero => + exact isInterlacingSeq0NonnegRealRooted_oneSupportSeq ⟨0, by lia⟩ + | succ n ih => + rw [anchorRefinedList_succ] + exact matrix_preserves_interlacing_seq0_of_2x2_weak + (anchorTransferMatrix d) + (anchorTransferMatrix_rect d) + (anchorTransferMatrix_nonneg d) + (anchorTransferMatrix_has2x2 d) + (anchorRefinedList d n) + (length_anchorRefinedList d n) + ih.1 ih.2 + +/-- Ordered refined components interlace. -/ +theorem anchorRefinedPolynomial_interl (d n : ℕ) (i j : Fin (d + 1)) + (hij : i < j) : + Interl (anchorRefinedPolynomial d n i) (anchorRefinedPolynomial d n j) := by + let i' : Fin (anchorRefinedList d n).length := + ⟨i, by rw [length_anchorRefinedList]; exact i.2⟩ + let j' : Fin (anchorRefinedList d n).length := + ⟨j, by rw [length_anchorRefinedList]; exact j.2⟩ + have hij' : i' < j' := by simpa [i', j'] using hij + have h := (anchorRefined_interlacing d n).1.1.interl hij' + simpa [anchorRefinedPolynomial, i', j'] using h + +/-- Every refined component is real-rooted, including the zero polynomial +under the library's splitting convention. -/ +theorem anchorRefinedPolynomial_splits (d n : ℕ) (i : Fin (d + 1)) : + (anchorRefinedPolynomial d n i).Splits := by + apply (anchorRefined_interlacing d n).splits_of_mem + exact List.get_mem _ _ + +/-- The unrefined anchor polynomial is the `∞` component one step later, as +in the paper's identity `Q_(d,n) = Q_(d,n+1,∞)`. -/ +def anchorRefinedTotal (d n : ℕ) : ℝ[X] := + anchorRefinedPolynomial d (n + 1) ⟨0, by lia⟩ + +/-- The total anchor polynomial obtained from the refined recursion is +real-rooted. -/ +theorem anchorRefinedTotal_splits (d n : ℕ) : + (anchorRefinedTotal d n).Splits := + anchorRefinedPolynomial_splits d (n + 1) ⟨0, by lia⟩ + +end + +end FerrersTiling +end RealRooted diff --git a/RealRooted/FerrersTiling/FaultFreeRectangle.lean b/RealRooted/FerrersTiling/FaultFreeRectangle.lean new file mode 100644 index 000000000..d16838063 --- /dev/null +++ b/RealRooted/FerrersTiling/FaultFreeRectangle.lean @@ -0,0 +1,96 @@ +import RealRooted.CombinatorialExamples.PathPowerClique + +/-! +# Fault-free two-column Ferrers-tile polynomials + +This file formalizes Ahlberg--Alexandersson, *Polynomials from tilings of +rectangles*, Lemmas 6.2 and 6.3, at the algebraic level. +-/ + +open Polynomial + +namespace RealRooted +namespace FerrersTiling + +noncomputable section + +/-- The fault-free polynomial `B_d` for the rectangular tile `(k,k)`. +Its coefficient of `X^m` is the bars-and-stars number from Lemma 6.2. -/ +def rectangleFaultFreePolynomial (k d : ℕ) : ℝ[X] := + pathPowerCliquePolynomial k (d + k - 1) + +/-- The closed coefficient formula from Ahlberg--Alexandersson, Lemma 6.2. -/ +@[simp] +theorem coeff_rectangleFaultFreePolynomial (k d m : ℕ) : + (rectangleFaultFreePolynomial k d).coeff m = + (Nat.choose (d + k - 1 - (k - 1) * m) m : ℝ) := by + exact coeff_pathPowerCliquePolynomial k (d + k - 1) m + +/-- The recurrence `B_d = B_(d-1) + X B_(d-k)` from +Ahlberg--Alexandersson, Lemma 6.3. -/ +theorem rectangleFaultFreePolynomial_eq_add (k d : ℕ) + (hk : 2 ≤ k) (hdk : k ≤ d) : + rectangleFaultFreePolynomial k d = + rectangleFaultFreePolynomial k (d - 1) + + X * rectangleFaultFreePolynomial k (d - k) := by + have h := pathPowerCliquePolynomial_add_lag k (d - 1) hk + simpa [rectangleFaultFreePolynomial, + show d - 1 + k = d + k - 1 by lia, + show d - 1 + k - 1 = (d - 1) + k - 1 by rfl, + show d - k + k - 1 = d - 1 by lia] using h + +/-- The initial values are `B_d = 1 + d X` when `d ≤ k`. -/ +theorem rectangleFaultFreePolynomial_eq_one_add (k d : ℕ) + (hk : 2 ≤ k) (hdk : d ≤ k) : + rectangleFaultFreePolynomial k d = 1 + C (d : ℝ) * X := by + ext m + rw [coeff_rectangleFaultFreePolynomial, coeff_add] + rcases m with _ | _ | m + · simp + · rw [coeff_one, coeff_C_mul, coeff_X] + norm_num + lia + · rw [coeff_one, coeff_C_mul, coeff_X] + simp only [Nat.succ_ne_zero, ite_false, zero_add] + norm_cast + apply Nat.choose_eq_zero_of_lt + have hm : 2 ≤ m + 2 := by lia + have hmul : 2 * k ≤ (m + 2) * k := Nat.mul_le_mul_right k hm + have hid : (k - 1) * (m + 2) + (m + 2) = (m + 2) * k := by + calc + (k - 1) * (m + 2) + (m + 2) = ((k - 1) + 1) * (m + 2) := by ring + _ = k * (m + 2) := by rw [Nat.sub_add_cancel (by lia)] + _ = (m + 2) * k := by ring + by_cases hle : (k - 1) * (m + 2) ≤ d + k - 1 + · apply (Nat.sub_lt_iff_lt_add hle).2 + have hbound : d + k - 1 < (m + 2) * k := + lt_of_lt_of_le (by lia) hmul + rw [← hid] at hbound + simpa [show m + 1 + 1 = m + 2 by lia, Nat.add_comm] using hbound + · rw [Nat.sub_eq_zero_of_le (le_of_not_ge hle)] + lia + +/-- Consecutive fault-free rectangle polynomials interlace. -/ +theorem rectangleFaultFreePolynomial_interl (k d : ℕ) + (hk : 2 ≤ k) (hd : 1 ≤ d) : + Interl (rectangleFaultFreePolynomial k (d - 1)) + (rectangleFaultFreePolynomial k d) := by + have h := strictInterl_pathPowerCliquePolynomial_add + k (d - 1 + k - 1) 1 hk (by norm_num) (by lia) + simpa [rectangleFaultFreePolynomial, + show d - 1 + k - 1 + 1 = d + k - 1 by lia] using h.toInterl + +/-- Every fault-free rectangle polynomial is Pólya-frequency, hence real-rooted. -/ +theorem rectangleFaultFreePolynomial_isPFPolynomial (k d : ℕ) : + IsPFPolynomial (rectangleFaultFreePolynomial k d) := + isPFPolynomial_pathPowerCliquePolynomial k (d + k - 1) + +/-- Every fault-free rectangle polynomial is real-rooted. -/ +theorem rectangleFaultFreePolynomial_splits (k d : ℕ) : + (rectangleFaultFreePolynomial k d).Splits := + splits_pathPowerCliquePolynomial k (d + k - 1) + +end + +end FerrersTiling +end RealRooted diff --git a/RealRooted/FerrersTiling/FaultLines.lean b/RealRooted/FerrersTiling/FaultLines.lean new file mode 100644 index 000000000..3325f9686 --- /dev/null +++ b/RealRooted/FerrersTiling/FaultLines.lean @@ -0,0 +1,89 @@ +import RealRooted.FerrersTiling.TwoColumn +import RealRooted.DegreeDropReversal + +/-! +# Fault-line polynomials for two-column Ferrers tiles + +This file formalizes the algebraic statement in Ahlberg--Alexandersson, +*Polynomials from tilings of rectangles*, Section 6: the fault-line family is +the coefficient reversal of the two-column family. +-/ + +open Polynomial + +namespace RealRooted +namespace FerrersTiling + +noncomputable section + +/-- The two-column polynomial has degree at most its board-width index. -/ +theorem twoColumnPolynomial_natDegree_le (d n : ℕ) : + (twoColumnPolynomial d n).natDegree ≤ n := by + have hbase := pathPowerCliquePolynomial_subsequence_natDegree_le + 2 1 0 n (by norm_num) (by norm_num) (by norm_num) + have hbase' : (pathPowerCliquePolynomial 2 n).natDegree ≤ n := by + simpa [pathPowerCliquePolynomial] using hbase + refine Polynomial.natDegree_comp_le.trans ?_ + have hinner : (C (d : ℝ) * X).natDegree ≤ 1 := by compute_degree + nlinarith [Nat.mul_le_mul hbase' hinner] + +/-- The fault-line polynomial `R_n = X^n P_n(X⁻¹)`. -/ +def faultLinePolynomial (d n : ℕ) : ℝ[X] := + (twoColumnPolynomial d n).reflect n + +@[simp] +theorem faultLinePolynomial_zero (d : ℕ) : faultLinePolynomial d 0 = 1 := by + simp [faultLinePolynomial, twoColumnPolynomial_zero] + +@[simp] +theorem faultLinePolynomial_one (d : ℕ) : faultLinePolynomial d 1 = X := by + simp [faultLinePolynomial, twoColumnPolynomial_one] + +private theorem reflect_succ_of_natDegree_le {p : ℝ[X]} {n : ℕ} + (hp : p.natDegree ≤ n) : + p.reflect (n + 1) = X * p.reflect n := by + have h := reflect_mul (1 : ℝ[X]) p (F := 1) (G := n) (by simp) hp + rw [one_mul, reflect_one, pow_one, add_comm] at h + exact h + +/-- The recurrence `R_(n+2) = X R_(n+1) + d X R_n`. -/ +theorem faultLinePolynomial_add_two (d n : ℕ) : + faultLinePolynomial d (n + 2) = + X * faultLinePolynomial d (n + 1) + + C (d : ℝ) * X * faultLinePolynomial d n := by + rw [faultLinePolynomial, twoColumnPolynomial_add_two] + rw [Polynomial.reflect_add] + rw [reflect_succ_of_natDegree_le (twoColumnPolynomial_natDegree_le d (n + 1))] + have hprod : + (C (d : ℝ) * X * twoColumnPolynomial d n).natDegree ≤ n + 1 := by + refine natDegree_mul_le.trans ?_ + have hleft : (C (d : ℝ) * X).natDegree ≤ 1 := by compute_degree + exact (add_le_add hleft (twoColumnPolynomial_natDegree_le d n)).trans_eq (by lia) + have hreflect : + (C (d : ℝ) * X * twoColumnPolynomial d n).reflect (n + 2) = + X * (C (d : ℝ) * X * twoColumnPolynomial d n).reflect (n + 1) := by + simpa only [show n + 2 = n + 1 + 1 by lia] using + reflect_succ_of_natDegree_le hprod + rw [hreflect] + have hmul := reflect_mul (C (d : ℝ) * X) (twoColumnPolynomial d n) + (F := 1) (G := n) (by compute_degree) (twoColumnPolynomial_natDegree_le d n) + have hmul' : + (C (d : ℝ) * X * twoColumnPolynomial d n).reflect (n + 1) = + (C (d : ℝ) * X).reflect 1 * (twoColumnPolynomial d n).reflect n := by + simpa only [Nat.add_comm] using hmul + rw [hmul'] + rw [show (C (d : ℝ) * X : ℝ[X]) = C (d : ℝ) * X ^ 1 by simp, + Polynomial.reflect_C_mul_X_pow, Polynomial.revAt_le (by lia)] + simp only [Nat.sub_self, pow_zero, mul_one, faultLinePolynomial] + ring + +/-- Every fault-line polynomial is real-rooted. -/ +theorem faultLinePolynomial_splits (d n : ℕ) : + (faultLinePolynomial d n).Splits := + Polynomial.Splits.reflect (twoColumnPolynomial_splits d n) + (twoColumnPolynomial_natDegree_le d n) + +end + +end FerrersTiling +end RealRooted diff --git a/RealRooted/FerrersTiling/LTiling.lean b/RealRooted/FerrersTiling/LTiling.lean new file mode 100644 index 000000000..3169f8160 --- /dev/null +++ b/RealRooted/FerrersTiling/LTiling.lean @@ -0,0 +1,74 @@ +import RealRooted.FerrersTiling.Rigid + +/-! +# Algebraic L-tile polynomials + +This file defines the unrefined polynomial family from +Ahlberg--Alexandersson, *Polynomials from tilings of rectangles*, Section 3, +by its closed form. Its later appearance as the rigid family with `ell = 1` +gives real-rootedness without using the layer-2 tiling model. +-/ + +open Polynomial + +namespace RealRooted +namespace FerrersTiling + +noncomputable section + +/-- The algebraic L-tile polynomial `P_(d,n)`. -/ +def lTilingPolynomial (d n : ℕ) : ℝ[X] := + rigidPolynomial d 1 n + +/-- The anchor-word polynomial `Q_(d,n)`, algebraically defined by the same +closed form in layer 1. -/ +def anchorPolynomial (d n : ℕ) : ℝ[X] := + rigidPolynomial d 1 n + +/-- The paper's polynomials `P_(d,n)` and `Q_(d,n)` agree. -/ +theorem lTilingPolynomial_eq_anchorPolynomial (d n : ℕ) : + lTilingPolynomial d n = anchorPolynomial d n := rfl + +/-- Closed coefficients of the L-tile polynomial. -/ +@[simp] +theorem coeff_lTilingPolynomial (d n j : ℕ) : + (lTilingPolynomial d n).coeff j = + (Nat.choose (d * (n - j)) j : ℝ) := by + simp [lTilingPolynomial] + +/-- L-tile polynomials are Pólya-frequency. -/ +theorem lTilingPolynomial_isPFPolynomial (d n : ℕ) : + IsPFPolynomial (lTilingPolynomial d n) := + rigidPolynomial_isPFPolynomial d 1 n + +/-- L-tile polynomials are real-rooted. -/ +theorem lTilingPolynomial_splits (d n : ℕ) : + (lTilingPolynomial d n).Splits := + rigidPolynomial_splits d 1 n + +/-- Every zero of an L-tile polynomial is nonpositive. -/ +theorem lTilingPolynomial_roots_nonpos (d n : ℕ) {r : ℝ} + (hr : r ∈ (lTilingPolynomial d n).roots) : r ≤ 0 := + (lTilingPolynomial_isPFPolynomial d n).roots_nonpos r hr + +/-- Consecutive L-tile polynomials interlace. -/ +theorem lTilingPolynomial_interl_succ (d n : ℕ) : + Interl (lTilingPolynomial d n) (lTilingPolynomial d (n + 1)) := by + by_cases hd : d = 0 + · subst d + have hzero (m : ℕ) : lTilingPolynomial 0 m = 1 := by + ext j + rw [coeff_lTilingPolynomial] + cases j with + | zero => simp + | succ j => rw [coeff_one]; simp + rw [hzero, hzero] + exact Interl.refl fun _ => Polynomial.Splits.one + · have h := strictInterl_pathPowerCliquePolynomial_add + (d + 1) (d * n) d (by lia) (Nat.pos_of_ne_zero hd) (by lia) + simpa [lTilingPolynomial, rigidPolynomial, Nat.mul_succ] using h.toInterl + +end + +end FerrersTiling +end RealRooted diff --git a/RealRooted/FerrersTiling/Rigid.lean b/RealRooted/FerrersTiling/Rigid.lean new file mode 100644 index 000000000..08c6c8b4d --- /dev/null +++ b/RealRooted/FerrersTiling/Rigid.lean @@ -0,0 +1,69 @@ +import RealRooted.CombinatorialExamples.PathPowerClique + +/-! +# Rigid Ferrers-tile polynomials + +This file formalizes the algebraic content of Ahlberg--Alexandersson, +*Polynomials from tilings of rectangles*, Theorem 5.3. The coefficient +formula is identified with an independence polynomial of a power of a path. +-/ + +open Finset Polynomial + +namespace RealRooted +namespace FerrersTiling + +noncomputable section + +/-- The rigid-tile polynomial with parameters `d`, tile width minus one `ell`, +and board width `n`. -/ +def rigidPolynomial (d ell n : ℕ) : ℝ[X] := + pathPowerCliquePolynomial (d * ell + 1) (d * n) + +/-- The coefficient formula +`[X^j] P = binom (d * (n - j * ell)) j` from +Ahlberg--Alexandersson, Theorem 5.3. -/ +@[simp] +theorem coeff_rigidPolynomial (d ell n j : ℕ) : + (rigidPolynomial d ell n).coeff j = + (Nat.choose (d * (n - j * ell)) j : ℝ) := by + rw [rigidPolynomial, coeff_pathPowerCliquePolynomial] + congr 2 + rw [show d * ell + 1 - 1 = d * ell by lia, + show d * ell * j = d * (j * ell) by ring, + ← Nat.mul_sub_left_distrib] + +/-- The closed form for rigid-tile polynomials. -/ +theorem rigidPolynomial_eq_sum (d ell n : ℕ) : + rigidPolynomial d ell n = + ∑ j ∈ range (d * n + 1), + C (Nat.choose (d * (n - j * ell)) j : ℝ) * X ^ j := by + ext j + rw [coeff_rigidPolynomial, finsetSum_coeff] + simp only [coeff_C_mul_X_pow, Finset.sum_ite_eq, mem_range] + split_ifs with hj + · rfl + · rw [Nat.choose_eq_zero_of_lt] + · simp + · exact lt_of_le_of_lt + (Nat.mul_le_mul_left d (Nat.sub_le n (j * ell))) (by lia) + +/-- Rigid-tile polynomials are Pólya-frequency polynomials. -/ +theorem rigidPolynomial_isPFPolynomial (d ell n : ℕ) : + IsPFPolynomial (rigidPolynomial d ell n) := + isPFPolynomial_pathPowerCliquePolynomial (d * ell + 1) (d * n) + +/-- Rigid-tile polynomials are real-rooted. -/ +theorem rigidPolynomial_splits (d ell n : ℕ) : + (rigidPolynomial d ell n).Splits := + splits_pathPowerCliquePolynomial (d * ell + 1) (d * n) + +/-- Every zero of a rigid-tile polynomial is nonpositive. -/ +theorem rigidPolynomial_roots_nonpos (d ell n : ℕ) {r : ℝ} + (hr : r ∈ (rigidPolynomial d ell n).roots) : r ≤ 0 := + (rigidPolynomial_isPFPolynomial d ell n).roots_nonpos r hr + +end + +end FerrersTiling +end RealRooted diff --git a/RealRooted/FerrersTiling/RowRefined.lean b/RealRooted/FerrersTiling/RowRefined.lean new file mode 100644 index 000000000..f2a7d0335 --- /dev/null +++ b/RealRooted/FerrersTiling/RowRefined.lean @@ -0,0 +1,45 @@ +import RealRooted.CombinatorialExamples.PathPowerClique + +/-! +# Row-refined Ferrers-tiling polynomials + +This file formalizes the univariate specialization in +Ahlberg--Alexandersson, *Polynomials from tilings of rectangles*, +Theorem 7.1. The multivariate tiling model is deferred to layer 2. +-/ + +open Finset Polynomial + +namespace RealRooted +namespace FerrersTiling + +noncomputable section + +/-- The specialization `S_((k,k),d,n)(s,1,...,1)`, defined by the closed +form in Ahlberg--Alexandersson, Theorem 7.1. -/ +def rowRefinedSpecialization (d n : ℕ) : ℝ[X] := + (pathPowerCliquePolynomial 2 n).comp (X + C ((d - 1 : ℕ) : ℝ)) + +/-- The closed form +`sum_j binom (n-j) j (s+d-1)^j` for the row-refined specialization. -/ +theorem rowRefinedSpecialization_eq_sum (d n : ℕ) : + rowRefinedSpecialization d n = + ∑ j ∈ range (n + 1), + C (Nat.choose (n - j) j : ℝ) * + (X + C ((d - 1 : ℕ) : ℝ)) ^ j := by + rw [rowRefinedSpecialization, pathPowerCliquePolynomial] + simp only [Polynomial.sum_comp, mul_comp, C_comp, pow_comp, X_comp] + congr 1 + funext j + norm_num + +/-- Every row-refined univariate specialization is real-rooted. -/ +theorem rowRefinedSpecialization_splits (d n : ℕ) : + (rowRefinedSpecialization d n).Splits := by + apply (splits_pathPowerCliquePolynomial 2 n).comp_of_natDegree_le_one + compute_degree + +end + +end FerrersTiling +end RealRooted diff --git a/RealRooted/FerrersTiling/TwoColumn.lean b/RealRooted/FerrersTiling/TwoColumn.lean new file mode 100644 index 000000000..bf76f3cd4 --- /dev/null +++ b/RealRooted/FerrersTiling/TwoColumn.lean @@ -0,0 +1,98 @@ +import RealRooted.CombinatorialExamples.PathPowerClique + +/-! +# Two-column Ferrers-tiling polynomials + +This file formalizes the algebraic content of Ahlberg--Alexandersson, +*Polynomials from tilings of rectangles*, Theorem 4.1. The combinatorial +identification with tilings is deliberately deferred: the family is defined +by its closed form. +-/ + +open Finset Polynomial + +namespace RealRooted.FerrersTiling + +noncomputable section + +/-- The two-column polynomial +`sum_j binom (n-j) j d^j X^j` from Ahlberg--Alexandersson, Theorem 4.1. -/ +def twoColumnPolynomial (d n : ℕ) : ℝ[X] := + (pathPowerCliquePolynomial 2 n).comp (C (d : ℝ) * X) + +/-- Coefficients of the two-column polynomial. -/ +@[simp] +theorem coeff_twoColumnPolynomial (d n j : ℕ) : + (twoColumnPolynomial d n).coeff j = + (Nat.choose (n - j) j : ℝ) * (d : ℝ) ^ j := by + rw [twoColumnPolynomial, Polynomial.comp_C_mul_X_coeff, + coeff_pathPowerCliquePolynomial] + norm_num + +/-- The closed form in Ahlberg--Alexandersson, Theorem 4.1. -/ +theorem twoColumnPolynomial_eq_sum (d n : ℕ) : + twoColumnPolynomial d n = + ∑ j ∈ range (n + 1), + C ((Nat.choose (n - j) j : ℝ) * (d : ℝ) ^ j) * X ^ j := by + ext j + rw [coeff_twoColumnPolynomial, finsetSum_coeff] + simp only [coeff_C_mul_X_pow, Finset.sum_ite_eq, mem_range] + split_ifs with hj + · rfl + · rw [Nat.choose_eq_zero_of_lt (by lia), Nat.cast_zero, zero_mul] + +@[simp] +theorem twoColumnPolynomial_zero (d : ℕ) : twoColumnPolynomial d 0 = 1 := by + ext j + rw [coeff_twoColumnPolynomial] + cases j with + | zero => simp + | succ j => rw [coeff_one]; simp + +@[simp] +theorem twoColumnPolynomial_one (d : ℕ) : twoColumnPolynomial d 1 = 1 := by + ext j + rw [coeff_twoColumnPolynomial] + cases j with + | zero => simp + | succ j => rw [coeff_one]; simp + +/-- The order-two recurrence in Ahlberg--Alexandersson, Theorem 4.1. -/ +theorem twoColumnPolynomial_add_two (d n : ℕ) : + twoColumnPolynomial d (n + 2) = + twoColumnPolynomial d (n + 1) + + C (d : ℝ) * X * twoColumnPolynomial d n := by + have h := congrArg (fun p : ℝ[X] ↦ p.comp (C (d : ℝ) * X)) + (pathPowerCliquePolynomial_add_lag 2 n (by norm_num)) + simpa [twoColumnPolynomial, mul_assoc] using h + +/-- Every two-column polynomial is real-rooted. -/ +theorem twoColumnPolynomial_splits (d n : ℕ) : + (twoColumnPolynomial d n).Splits := by + apply (splits_pathPowerCliquePolynomial 2 n).comp_of_natDegree_le_one + by_cases hd : d = 0 + · simp [hd] + · simp [hd] + +/-- Consecutive two-column polynomials interlace. -/ +theorem twoColumnPolynomial_interl_succ (d n : ℕ) : + Interl (twoColumnPolynomial d n) (twoColumnPolynomial d (n + 1)) := by + by_cases hd : d = 0 + · subst d + have hzero (m : ℕ) : twoColumnPolynomial 0 m = 1 := by + ext j + rw [coeff_twoColumnPolynomial] + cases j with + | zero => simp + | succ j => rw [coeff_one]; simp + rw [hzero, hzero] + exact Interl.refl fun _ => Polynomial.Splits.one + · exact (strictInterl_pathPowerCliquePolynomial_add 2 n 1 + (by norm_num) (by norm_num) (by norm_num)).toInterl.comp_C_mul_X + (by exact_mod_cast Nat.pos_of_ne_zero hd) + +end + +end FerrersTiling + +end RealRooted diff --git a/RealRooted/Production.lean b/RealRooted/Production.lean index 248f88df8..ce30c5d8e 100644 --- a/RealRooted/Production.lean +++ b/RealRooted/Production.lean @@ -1882,6 +1882,14 @@ import RealRooted.Mathlib.Algebra.LinearRecurrence.Operator import RealRooted.RecurrenceSpace.Defect import RealRooted.RecurrenceSpace.Filtered import RealRooted.RecurrenceSpace.Operators +import RealRooted.FerrersTiling.AnchorRefined +import RealRooted.FerrersTiling.FaultFreeRectangle +import RealRooted.FerrersTiling.FaultLines +import RealRooted.FerrersTiling.LTiling +import RealRooted.FerrersTiling.Rigid +import RealRooted.FerrersTiling.RowRefined +import RealRooted.FerrersTiling.TwoColumn +import RealRooted.Challenges.FerrersTilings /-! # RealRooted production umbrella diff --git a/scripts/import_architecture.json b/scripts/import_architecture.json index fb4d36acd..4acebf8dd 100644 --- a/scripts/import_architecture.json +++ b/scripts/import_architecture.json @@ -16,10 +16,10 @@ ], "budgets": { "RealRooted": { - "max_modules": 2056 + "max_modules": 2064 }, "RealRooted.Production": { - "max_modules": 1877 + "max_modules": 1885 }, "RealRooted.Tactic.Examples": { "max_modules": 931 diff --git a/website/symcat/bibliography.json b/website/symcat/bibliography.json index 4793ba669..3b2616c2d 100644 --- a/website/symcat/bibliography.json +++ b/website/symcat/bibliography.json @@ -1,5 +1,31 @@ { "entries": { + "AhlbergAlexandersson2026": { + "bibtex": "@article{AhlbergAlexandersson2026,\n author = {John Ahlberg and Per Alexandersson},\n title = {Polynomials from tilings of rectangles},\n year = {2026},\n eprint = {2605.03473},\n archivePrefix = {arXiv},\n primaryClass = {math.CO}\n}\n", + "csl": { + "URL": "https://arxiv.org/abs/2605.03473", + "author": [ + { + "family": "Ahlberg", + "given": "John" + }, + { + "family": "Alexandersson", + "given": "Per" + } + ], + "issued": { + "date-parts": [ + [ + 2026 + ] + ] + }, + "title": "Polynomials from tilings of rectangles", + "type": "article-journal" + }, + "label": "AA26" + }, "AissenSchoenbergWhitney1952": { "bibtex": "@article{AissenSchoenbergWhitney1952,\n author = {Aissen, Michael and Schoenberg, I. J. and Whitney, A. M.},\n title = {On the generating functions of totally positive sequences {I}},\n year = {1952},\n journal = {Journal d'Analyse Mathématique},\n volume = {2},\n number = {1},\n pages = {93--103},\n doi = {10.1007/BF02786970},\n url = {https://doi.org/10.1007/BF02786970}\n}\n", "csl": { diff --git a/website/symcat/labels.json b/website/symcat/labels.json index 6bc4e0b25..8bb7532a3 100644 --- a/website/symcat/labels.json +++ b/website/symcat/labels.json @@ -1 +1 @@ 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ope","findstatPermutationCSP":"cyclic-sieving.htm#findstatPermutationCSP","finiteMultiplierSequenceCriterion":"realRooted.htm#finiteMultiplierSequenceCriterion","finitePolyaFrequencySequence":"polyaFrequency.htm#finitePolyaFrequencySequence","firstDerivativeCriterion":"polynomialRecurrences.htm#firstDerivativeCriterion","fixedPointFreeStanleySymmetricFunction":"stanleySymmetric.htm#fixedPointFreeStanleySymmetricFunction","flagMatroid":"matroids.htm#flagMatroid","flagVarieties":"varieties.htm#flagVarieties","flagVarietyDefinition":"varieties.htm#flagVarietyDefinition","flaggedFactorialGrothendieckPolynomial":"grothendieck.htm#flaggedFactorialGrothendieckPolynomial","flaggedKostkaCoefficientDefinition":"schurFlagged.htm#flaggedKostkaCoefficientDefinition","flaggedLLT":"llt.htm#flaggedLLT","flaggedLLTPolynomialDefinition":"llt.htm#flaggedLLTPolynomialDefinition","flaggedPPartition":"pPartitions.htm#flaggedPPartition","flaggedRefinedSkewStableGrothendieck":"grothendieck.htm#flaggedRefinedSkewStableGrothendieck","flaggedSchurDividedDifferences":"schurFlagged.htm#flaggedSchurDividedDifferences","flaggedSchurKeyExpansion":"schurFlagged.htm#flaggedSchurKeyExpansion","flaggedSchurKostka":"schurFlagged.htm#flaggedSchurKostka","flaggedSchurPolynomialDefinition":"schurFlagged.htm#flaggedSchurPolynomialDefinition","flaggedSchurPolynomialJacobiTrudi":"schurFlagged.htm#flaggedSchurPolynomialJacobiTrudi","flaggedSchurRootsOfUnity":"schurFlagged.htm#flaggedSchurRootsOfUnity","flaggedSetValuedTableau":"grothendieck.htm#flaggedSetValuedTableau","flaggedSkewSchurPolynomialDefinition":"schurFlagged.htm#flaggedSkewSchurPolynomialDefinition","flaggedWeylModule":"schubert.htm#flaggedWeylModule","flatComposition":"assaf.htm#flatComposition","flippedExtSchur":"qsymSchur.htm#flippedExtSchur","flippedExtendedSchurFunction":"qsymSchur.htm#flippedExtendedSchurFunction","flowPolytope":"polytopes.htm#flowPolytope","flowPolytopeDefinition":"polytopes.htm#flowPolytopeDefinition","foataStrehlAction":"gammaPositivity.htm#foataStrehlAction","fockSpace":"symmetricFunctionsSpace.htm#fockSpace","fominKirillovConjecture":"schubert.htm#fominKirillovConjecture","forestPolynomialDefinition":"assaf.htm#forestPolynomialDefinition","forestPolynomials":"assaf.htm#forestPolynomials","forgotten":"standardSymmetricFunctions.htm#forgotten","foulkesConjecture":"plethysm.htm#foulkesConjecture","frobeniusCharacteristic":"representationTheory.htm#frobeniusCharacteristic","frobeniusCoordinates":"preliminaries.htm#frobeniusCoordinates","frobeniusImage":"representationTheory.htm#frobeniusImage","fullyInterlacingLace":"realRootedInterlacing.htm#fullyInterlacingLace","fullyInterlacingSequence":"realRootedInterlacing.htm#fullyInterlacingSequence","fultonsConjecture":"littlewoodRichardson.htm#fultonsConjecture","fundamentalParticle":"assaf.htm#fundamentalParticle","fundamentalParticleBasis":"assaf.htm#fundamentalParticleBasis","fundamentalSlidePolynomialDefinition":"assaf.htm#fundamentalSlidePolynomialDefinition","fusionCoefficient":"cylindricSchur.htm#fusionCoefficient","fussCatalanObjects":"combinatorialObjects.htm#fussCatalanObjects","galConjecture":"gammaPositivity.htm#galConjecture","gammaPolynomial":"gammaPositivity.htm#gammaPolynomial","gammaPositiveExamples":"gammaPositivity.htm#gammaPositiveExamples","gammaPositivePolynomial":"gammaPositivity.htm#gammaPositivePolynomial","gammaPositivity":"gammaPositivity.htm#gammaPositivity","gammaPositivityMethods":"gammaPositivity.htm#gammaPositivityMethods","gardingPolynomial":"stablePolynomials.htm#gardingPolynomial","gardingPolynomials":"stablePolynomials.htm#gardingPolynomials","generalLinearGroup":"representationTheory.htm#generalLinearGroup","generalUsefulThings":"resources.htm#generalUsefulThings","generalizedDeltaSquareConjecture":"shuffleAndDelta.htm#generalizedDeltaSquareConjecture","generalizedDemazureAtoms":"generalizedDemazureAtoms.htm#generalizedDemazureAtoms","generalizedDualStableGrothendieckPolynomial":"grothendieck.htm#generalizedDualStableGrothendieckPolynomial","generalizedFactorialSchurFunction":"schurShifted.htm#generalizedFactorialSchurFunction","generalizedFlaggedSchurModule":"representationTheory.htm#generalizedFlaggedSchurModule","generalizedNarayanaPolynomialsRealRooted":"realRootedCatalan.htm#generalizedNarayanaPolynomialsRealRooted","generalizedNets":"chromaticQuasisymmetric.htm#generalizedNets","generalizedPermutohedron":"polytopes.htm#generalizedPermutohedron","generalizedPermutohedronDefinition":"polytopes.htm#generalizedPermutohedronDefinition","generalizedPitmanStanleyPolytope":"polytopes.htm#generalizedPitmanStanleyPolytope","generalizedSchurP":"schurPQ.htm#generalizedSchurP","generalizedSchurPFunction":"schurPQ.htm#generalizedSchurPFunction","generalizedSchurQ":"schurPQ.htm#generalizedSchurQ","generalizedSchurQFunction":"schurPQ.htm#generalizedSchurQFunction","generalizedSnakePoset":"realRooted.htm#generalizedSnakePoset","generalizedSnakeRealRooted":"realRooted.htm#generalizedSnakeRealRoo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iordanArrayPF":"polyaFrequency.htm#riordanArrayPF","riseDeltaConjecture":"shuffleAndDelta.htm#riseDeltaConjecture","roichmanChi":"murnaghanNakayama.htm#roichmanChi","rookLatticePathMatroids":"lattice-path-matroids.htm#rookLatticePathMatroids","rookLatticePathPolytopes":"lattice-path-matroids.htm#rookLatticePathPolytopes","rookMatroidDefinition":"lattice-path-matroids.htm#rookMatroidDefinition","rookMatroidPolynomials":"lattice-path-matroids.htm#rookMatroidPolynomials","rookMatroids":"lattice-path-matroids.htm#rookMatroids","rookNesting":"lattice-path-matroids.htm#rookNesting","rookPolynomial":"realRootedGraphs.htm#rookPolynomial","rookPolynomials":"realRootedGraphs.htm#rookPolynomials","rookTheory":"chromaticQuasisymmetric.htm#rookTheory","rootIdeal":"kschur.htm#rootIdeal","rootOrder":"realRootedInterlacing.htm#rootOrder","rootPolytope":"polytopes.htm#rootPolytope","rootPolytopeDefinition":"polytopes.htm#rootPolytopeDefinition","rootSystem":"root-systems.htm#rootSystem","rootSystemClassification":"root-systems.htm#rootSystemClassification","rootedLabeledTreeDefinition":"parking-functions.htm#rootedLabeledTreeDefinition","rootedTreePoset":"posets.htm#rootedTreePoset","rotheDiagram":"permutations.htm#rotheDiagram","rotheDiagramDefinition":"permutations.htm#rotheDiagramDefinition","rowGeneratingFunction":"realRooted.htm#rowGeneratingFunction","rowInsertion":"rsk.htm#rowInsertion","rowStrictDualImmaculateSchurFunction":"qsymSchur.htm#rowStrictDualImmaculateSchurFunction","rowStrictExtendedSchurFunction":"qsymSchur.htm#rowStrictExtendedSchurFunction","rowStrictImmaculateTableau":"qsymSchur.htm#rowStrictImmaculateTableau","rowStrictQuasisymmetricSchurFunction":"qsymSchur.htm#rowStrictQuasisymmetricSchurFunction","rowStrictYoungQuasisymmetricSchurFunction":"qsymSchur.htm#rowStrictYoungQuasisymmetricSchurFunction","rowmotion":"rowmotionPromotion.htm#rowmotion","rowmotionPromotion":"rowmotionPromotion.htm#rowmotionPromotion","rowmotionPromotionConjugacy":"rowmotionPromotion.htm#rowmotionPromotionConjugacy","rowmotionTamari":"rowmotionPromotion.htm#rowmotionTamari","rsExtSchur":"qsymSchur.htm#rsExtSchur","rsdiSchur":"qsymSchur.htm#rsdiSchur","rsk":"rsk.htm#rsk","rskBurge":"rsk.htm#rskBurge","rskCorrespondence":"rsk.htm#rskCorrespondence","rskDual":"rsk.htm#rskDual","rskOne":"rsk.htm#rskOne","rskProperties":"rsk.htm#rskProperties","rskStandardization":"rsk.htm#rskStandardization","rskYangBaxter":"rsk.htm#rskYangBaxter","rsqSchur":"qsymSchur.htm#rsqSchur","rsyqSchur":"qsymSchur.htm#rsyqSchur","runSortedDescentPolynomial":"realRootedWords.htm#runSortedDescentPolynomial","sEulerianPolynomial":"realRootedWords.htm#sEulerianPolynomial","sPerpTrick":"plethysm.htm#sPerpTrick","samePhaseCompatible":"stablePolynomials.htm#samePhaseCompatible","samePhaseStabilityDefinition":"stablePolynomials.htm#samePhaseStabilityDefinition","samePhaseStable":"stablePolynomials.htm#samePhaseStable","schedule":"shuffleAndDelta.htm#schedule","schubert":"schubert.htm#schubert","schubertAffine":"schubertVariations.htm#schubertAffine","schubertAsSumOfElementary":"schubert.htm#schubertAsSumOfElementary","schubertBCD":"schubertVariations.htm#schubertBCD","schubertBackStable":"backStableSchubert.htm#schubertBackStable","schubertBackStableDouble":"schubertVariations.htm#schubertBackStableDouble","schubertBlockDecomposition":"schubert.htm#schubertBlockDecomposition","schubertBumplessPipeDream":"schubert.htm#schubertBumplessPipeDream","schubertCalculus":"schubert-calculus.htm#schubertCalculus","schubertCalculusComputationRules":"schubert-calculus.htm#schubertCalculusComputationRules","schubertCalculusDefinition":"schubert-calculus.htm#schubertCalculusDefinition","schubertCalculusFlagVariety":"schubert-calculus.htm#schubertCalculusFlagVariety","schubertCalculusGiambelli":"schubert-calculus.htm#schubertCalculusGiambelli","schubertCalculusGr24":"schubert-calculus.htm#schubertCalculusGr24","schubertCalculusLRTruncation":"schubert-calculus.htm#schubertCalculusLRTruncation","schubertCalculusPieri":"schubert-calculus.htm#schubertCalculusPieri","schubertCalculusPoincareDuality":"schubert-calculus.htm#schubertCalculusPoincareDuality","schubertCauchyFormula":"schubert.htm#schubertCauchyFormula","schubertCharge":"various-research.htm#schubertCharge","schubertComplexity":"schubert.htm#schubertComplexity","schubertDouble":"schubertVariations.htm#schubertDouble","schubertDoubleGiambelliFormula":"schubertVariations.htm#schubertDoubleGiambelliFormula","schubertDual":"schubertVariations.htm#schubertDual","schubertEnriched":"schubertVariations.htm#schubertEnriched","schubertGTFormula":"schubert.htm#schubertGTFormula","schubertInvolution":"schubertVariations.htm#schubertInvolution","schubertKeyExpansion":"schubert.htm#schubertKeyExpansion","schubertLittlewoodRichardson":"schubert.htm#schubertLittlewoodRichardson","schubertMatroidDefinition":"varieties.htm#schubertMatroidDefinition","schubertMonksRule":"schubert.htm#schubertMonksRule","schubertMurnaghanNakayama":"schubert.h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ope","findstatPermutationCSP":"cyclic-sieving.htm#findstatPermutationCSP","finiteMultiplierSequenceCriterion":"realRooted.htm#finiteMultiplierSequenceCriterion","finitePolyaFrequencySequence":"polyaFrequency.htm#finitePolyaFrequencySequence","firstDerivativeCriterion":"polynomialRecurrences.htm#firstDerivativeCriterion","fixedPointFreeStanleySymmetricFunction":"stanleySymmetric.htm#fixedPointFreeStanleySymmetricFunction","flagMatroid":"matroids.htm#flagMatroid","flagVarieties":"varieties.htm#flagVarieties","flagVarietyDefinition":"varieties.htm#flagVarietyDefinition","flaggedFactorialGrothendieckPolynomial":"grothendieck.htm#flaggedFactorialGrothendieckPolynomial","flaggedKostkaCoefficientDefinition":"schurFlagged.htm#flaggedKostkaCoefficientDefinition","flaggedLLT":"llt.htm#flaggedLLT","flaggedLLTPolynomialDefinition":"llt.htm#flaggedLLTPolynomialDefinition","flaggedPPartition":"pPartitions.htm#flaggedPPartition","flaggedRefinedSkewStableGrothendieck":"grothendieck.htm#flaggedRefinedSkewStableGrothendieck","flaggedSchurDividedDifferences":"schurFlagged.htm#flaggedSchurDividedDifferences","flaggedSchurKeyExpansion":"schurFlagged.htm#flaggedSchurKeyExpansion","flaggedSchurKostka":"schurFlagged.htm#flaggedSchurKostka","flaggedSchurPolynomialDefinition":"schurFlagged.htm#flaggedSchurPolynomialDefinition","flaggedSchurPolynomialJacobiTrudi":"schurFlagged.htm#flaggedSchurPolynomialJacobiTrudi","flaggedSchurRootsOfUnity":"schurFlagged.htm#flaggedSchurRootsOfUnity","flaggedSetValuedTableau":"grothendieck.htm#flaggedSetValuedTableau","flaggedSkewSchurPolynomialDefinition":"schurFlagged.htm#flaggedSkewSchurPolynomialDefinition","flaggedWeylModule":"schubert.htm#flaggedWeylModule","flatComposition":"assaf.htm#flatComposition","flippedExtSchur":"qsymSchur.htm#flippedExtSchur","flippedExtendedSchurFunction":"qsymSchur.htm#flippedExtendedSchurFunction","flowPolytope":"polytopes.htm#flowPolytope","flowPolytopeDefinition":"polytopes.htm#flowPolytopeDefinition","foataStrehlAction":"gammaPositivity.htm#foataStrehlAction","fockSpace":"symmetricFunctionsSpace.htm#fockSpace","fominKirillovConjecture":"schubert.htm#fominKirillovConjecture","forestPolynomialDefinition":"assaf.htm#forestPolynomialDefinition","forestPolynomials":"assaf.htm#forestPolynomials","forgotten":"standardSymmetricFunctions.htm#forgotten","foulkesConjecture":"plethysm.htm#foulkesConjecture","frobeniusCharacteristic":"representationTheory.htm#frobeniusCharacteristic","frobeniusCoordinates":"preliminaries.htm#frobeniusCoordinates","frobeniusImage":"representationTheory.htm#frobeniusImage","fullyInterlacingLace":"realRootedInterlacing.htm#fullyInterlacingLace","fullyInterlacingSequence":"realRootedInterlacing.htm#fullyInterlacingSequence","fultonsConjecture":"littlewoodRichardson.htm#fultonsConjecture","fundamentalParticle":"assaf.htm#fundamentalParticle","fundamentalParticleBasis":"assaf.htm#fundamentalParticleBasis","fundamentalSlidePolynomialDefinition":"assaf.htm#fundamentalSlidePolynomialDefinition","fusionCoefficient":"cylindricSchur.htm#fusionCoefficient","fussCatalanObjects":"combinatorialObjects.htm#fussCatalanObjects","galConjecture":"gammaPositivity.htm#galConjecture","gammaPolynomial":"gammaPositivity.htm#gammaPolynomial","gammaPositiveExamples":"gammaPositivity.htm#gammaPositiveExamples","gammaPositivePolynomial":"gammaPositivity.htm#gammaPositivePolynomial","gammaPositivity":"gammaPositivity.htm#gammaPositivity","gammaPositivityMethods":"gammaPositivity.htm#gammaPositivityMethods","gardingPolynomial":"stablePolynomials.htm#gardingPolynomial","gardingPolynomials":"stablePolynomials.htm#gardingPolynomials","generalLinearGroup":"representationTheory.htm#generalLinearGroup","generalUsefulThings":"resources.htm#generalUsefulThings","generalizedDeltaSquareConjecture":"shuffleAndDelta.htm#generalizedDeltaSquareConjecture","generalizedDemazureAtoms":"generalizedDemazureAtoms.htm#generalizedDemazureAtoms","generalizedDualStableGrothendieckPolynomial":"grothendieck.htm#generalizedDualStableGrothendieckPolynomial","generalizedFactorialSchurFunction":"schurShifted.htm#generalizedFactorialSchurFunction","generalizedFlaggedSchurModule":"representationTheory.htm#generalizedFlaggedSchurModule","generalizedNarayanaPolynomialsRealRooted":"realRootedCatalan.htm#generalizedNarayanaPolynomialsRealRooted","generalizedNets":"chromaticQuasisymmetric.htm#generalizedNets","generalizedPermutohedron":"polytopes.htm#generalizedPermutohedron","generalizedPermutohedronDefinition":"polytopes.htm#generalizedPermutohedronDefinition","generalizedPitmanStanleyPolytope":"polytopes.htm#generalizedPitmanStanleyPolytope","generalizedSchurP":"schurPQ.htm#generalizedSchurP","generalizedSchurPFunction":"schurPQ.htm#generalizedSchurPFunction","generalizedSchurQ":"schurPQ.htm#generalizedSchurQ","generalizedSchurQFunction":"schurPQ.htm#generalizedSchurQFunction","generalizedSnakePoset":"realRooted.htm#generalizedSnakePoset","generalizedSnakeRealRooted":"realRooted.htm#generalizedSnakeRealRoo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